Pressure distribution in a fluid Pressure and pressure gradient Lecture 4 Mecânica de Fluidos Ambiental 2015/2016

Mecânica de Fluidos Ambiental 2015/2016

Pressure distribution in a fluidPressure and pressure gradient

Lecture 4


Chapter 2

• Concept of pressure and pressure gradient• Hydrostatic pressure distribution• Hydrostatic forces• Buoyance and stability


Forces in fluids

• Surface or volumetric (or mass)• Surface forces can be normal (pressure) or

tangential (friction)• Friction Forces are always parallel to velocity. This

is why they are often called “tangential forces”. Their equation can be complex if velocity is not parallel to any reference axis.


Pressure

• Many fluid problems do not involve motion. They concern the pressure distribution in a static fluid and its effect on solid surfaces and on floating and submerged bodies.

• Fluids at rest cannot support shear stress.• Pressure is used to indicate the normal force per

unit of area at a given point acting on a given plane within the fluid mass of interest.

• How the pressure at a point varies with the orientation of the plane passing through the point?


Pressure at a point

• Small wedge (“pequena cunha”) of fluid at rest of size x by z by s and depth b into the paper.

• There is no shear by definition, but we postulate that the pressures px, pz, and pn may be different on each face

• The weight of the element also may be important. • The element is assumed small, so the pressure is constant on each face. • Summation of forces must equal zero (no acceleration) in both the x and z directions.


Pressure at a point

By the geometry of the wedge:

These relations illustrate two important principles of the hydrostatic, or shear-free, condition: (1) there is no pressure change in the horizontal direction(2) there is a vertical change in pressure proportional to the density, gravity, and depth change

In the limit as the fluid wedge shrinks to a “point”, :

Since is arbitrary, we conclude that the pressure at point (p) in a static fluid is a point property, independent of the plane orientation.


Pressure force in fluid element• Pressure (or any other stress, for that matter) causes a

net force on a fluid element when it varies spatially.• Consider the pressure acting on the two x faces in Fig2.2 and let

the pressure vary arbitrarily:

• The net force in x direction on the element is:

• The total net-force vector on the element due to pressure is:


Equilibrium of fluid element• Forces acting on the fluid element: surface forces due to the pressure,

body force equal to weight of the element and friction

By volume unit and considering constante:

Weight=g(dxdydz)

(𝜏 𝑦𝑥 )𝑦≈(𝜇 𝑑𝑉𝑑𝑦

)𝑦

(𝜏 𝑦𝑥 )𝑦+𝑑𝑦≈(𝜇 𝑑𝑉𝑑𝑦

)𝑦+𝑑𝑦


Equilibrium of fluid element

• (net pressure force per unit volume)• (net gravity force per unit volume)• (net viscous force per unit volume)

• By Newton’s law, the sum of these per-unit-volume forces equals the mass per unit volume (density) times the acceleration a of the fluid element:


General equation

• If the fluid is at rest or at constant velocity, a=0 and the equation for the pressure distribution reduces to

This is a hydrostatic distribution and is correct for all fluids at rest, regardless of their viscosity, because the viscous term vanishes identically.


Hydrostatic pressure

• In our customary coordinate system z is “up”. Thus the local-gravity vector for small-scale problems is:

where g is the magnitude of local gravity, for example, 9.807 m/s2.

For these coordinates, equation has the components

• These equations show that pressure does not depend on x and y. Thus, as we move point to point in horizontal plane, pressure do not change. Since p depends only on z, can be replaced by the total derivative


Hydrostatic pressure

or

• These equation is the solution to the hydrostatic problem. The integration requires an assumption about the density and gravity distribution. Gases and liquids are usually treated differently.

• Pressure in a continuously distributed uniform static fluid varies only with vertical distance and is independent of the shape of the container.

• The pressure is the same at all points on a given horizontal plane in the fluid.

• The pressure increases with depth in the fluid.


Efect of variable gravity

• For a spherical planet of uniform density, the acceleration of gravity varies inversely as the square of the radius from its center

where r0 is the planet radius and g0 is the surface value of g. For earth, r0 3960 statute mi 6400 km.

In typical engineering problems the deviation from r0 extends from the deepest ocean, about 11 km, to the atmospheric height of supersonic transport operation, about 20 km. This gives a maximum variation in g of (6400/6420)2, or 0.6 percent. We therefore neglect the variation of g in most problems.


Hydrostatic pressure in liquids

• Liquids are so nearly incompressible that we can neglect their density variation in hydrostatics. Thus we assume constant density in liquid hydrostatic calculations, for which hydrostatic conditions integrates to

or

We use the first form in most problems. The quantity is called the specific weight (“peso específico”) of the fluid, with dimensions of weight per unit volume; The quantity is a length called the pressure head (“carga de pressão”) of the fluid.


Hidrostatic pressure distribution

• For lakes and oceans, the coordinate system is usually chosen as in Figure, with z=0 at the free surface, where p equals the surface atmospheric pressure pa.


Hydrostatic pressure in gases• Gases are compressible, with density nearly proportional to

pressure. Thus density must be considered as a variable in the hydrostatic pressure equation. It is sufficiently accurate to introduce the perfect-gas law in the hydrostatic equation:

Separate the variables and integrate between points 1 and 2:

The integral over z requires an assumption about the temperature variation T(z). One common approximation is the isothermal atmosphere, where T= T0:


Hydrostatic forces on plane surfaces

• Objective: compute the hydrostatic force over flat and curve surfaces and the application point (center of pressure).

• When a surface is submerged in a fluid, forces develop on the surface due to the fluid. The determination of these forces is important in the design of storage tanks, ships, dams, and other hydraulic structures.


Hydrostatic forces on plane surfaces• Plane panel of arbitrary shape completely submerged in a liquid• Panel plane makes an arbitrary angle with the horizontal free

surface, so that the depth varies over the panel surface.

•If h is the depth to any element area dA of the plate, from hydrostatic pressure equation in liquids the pressure there is p=pa+h.


Hydrostatic forces on plane surfaces• To derive formulas involving the plate shape, establish an xy

coordinate system in the plane of the plate with the origin at its centroid, plus a dummy coordinate down from the surface in the plane of the plate. A is the area of the plane surface.

• The total hydrostatic force on one side of the plate is given by

From the figure and, by definition, the centroidal slant distancefrom the surface to the plate is


• Therefore, since is constant along the plate, becomes

• Finally, unravel this by noticing that , the depth straight down from the surface to the plate centroid. Thus

• The force on one side of any plane submerged surface in a uniform fluid equals the pressure at the plate centroid times the plate area, independent of the shape of the plate or the angle at which it is slanted.


Center of pressure• However, to balance the bending-moment portion of the stress, the

resultant force F acts not through the centroid but below it toward the high-pressure side. Its line of action passes through the center of pressure CP of the plate (see figure).

• To find the coordinates (xCP, yCP), we sum moments of the elemental force p dA about the centroid and equate to the moment of the resultant F. To compute yCP, we equate

Ixx is the area moment of inertia of the plate area about its centroidal x axis

Vanishes by def. of centroidal axes


Center of pressure• The negative sign shows that yCP is below the

centroid at a deeper level (below the gravity center) and, unlike F, depends on angle .

• The determination of xCP is exactly similar:

Ixythe product of inertia of the plate, again computed in the plane of the plate

• If Ixy = 0, usually implying symmetry, xCP= 0 and the center of pressure lies directly below the centroid on the y axis.


• In most cases the ambient pressure pa is neglected because it acts on both sides of the plate; for example, the other side of the plate is inside a ship or on the dry side of a gate or dam. In this case pCG=hCG, and the center of pressure becomes independent of specific weight:


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Pressure distribution in a fluid Pressure and pressure gradient Lecture 4 Mecânica de Fluidos Ambiental 2015/2016